scholarly journals Farthest Points and Subdifferential in -Normed Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
S. Hejazian ◽  
A. Niknam ◽  
S. Shadkam
Keyword(s):  
Compact Subset ◽  
Normed Space ◽  
Dense Subset ◽  
Normed Spaces ◽  
Point Mapping ◽  
Farthest Points ◽  
Farthest Point ◽  
In Virtue Of

We study the farthest point mapping in a -normed space in virtue of subdifferential of , where is a weakly sequentially compact subset of . We show that the set of all points in which have farthest point in contains a dense subset of .

scholarly journals Asymptotic farthest points and extreme points

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5875-5885
Author(s):  
Nejhad Ardakani ◽  
Mazaheri Tehrani
Keyword(s):  
Functional Analysis ◽  
Approximation Theory ◽  
Normed Space ◽  
Extreme Points ◽  
Bounded Sequence ◽  
Bounded Subset ◽  
Farthest Points ◽  
Farthest Point

Let X be a normed space, G a nonempty bounded subset of X and fxng a bounded sequence in X. In this article, we introduce and discuss the concept of asymptotic farthest points of fxng in G, which is a new definition in abstract approximation theory. Then, by applying the topics of functional analysis, we investigate the relation between this new concept and the concepts of extreme points and convexity. In particular, one of the main purposes of this paper is to study conditions under which the existence (uniqueness) of asymptotic farthest point of fxng in G is equivalent to the existence (uniqueness) of asymptotic farthest point of fxng in ext(G) or co(G).

scholarly journals Properties of typical bounded closed convex sets in Hilbert space

2005 ◽  
Vol 2005 (4) ◽  
pp. 423-436
Author(s):  
F. S. de Blasi ◽  
N. V. Zhivkov
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Convex Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Dense Subset ◽  
Baire Category ◽  
Point Mapping ◽  
Farthest Point ◽  
Closed Convex Set

For a nonempty separable convex subsetXof a Hilbert spaceℍ(Ω), it is typical (in the sense of Baire category) that a bounded closed convex setC⊂ℍ(Ω)defines anm-valued metric antiprojection (farthest point mapping) at the points of a dense subset ofX, whenevermis a positive integer such thatm≤dimX+1.

scholarly journals A farthest-point characterisation of the relative Chebyshev centre

1996 ◽  
Vol 54 (1) ◽  
pp. 27-33 ◽  
Author(s):  
R. Huotari ◽  
M.P. Prophet ◽  
J. Shi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Hull ◽  
Compact Subset ◽  
Real Banach Space ◽  
Metric Projection ◽  
Closed Convex Hull ◽  
Farthest Points ◽  
Farthest Point ◽  
Gateaux Derivative

We characterise the relative Chebyshev centre of a compact subsetFof a real Banach space in terms of the Gateaux derivative of the distance to farthest points. We present a relative-Chebyshev-centre characterisation of Hilbert space. In Hilbert space we show that the relative Chebyshev centre is in the closed convex hull of the metric projection ofF, and we estimate the relative Chebyshev radius ofF.

scholarly journals Farthest points and monotone operators

1998 ◽  
Vol 58 (1) ◽  
pp. 75-92 ◽  
Author(s):  
U. Westphal ◽  
T. Schwartz
Keyword(s):  
Banach Spaces ◽  
Distance Function ◽  
Hilbert Spaces ◽  
Baire Category ◽  
Monotone Operators ◽  
New Approach ◽  
Point Mapping ◽  
Farthest Points ◽  
Intimate Connection ◽  
Farthest Point

We apply the theory of monotone operators to study farthest points in closed bounded subsets of real Banach spaces. This new approach reveals the intimate connection between the farthest point mapping and the subdifferential of the farthest distance function. Moreover, we prove that a typical exception set in the Baire category sense is pathwise connected. Stronger results are obtained in Hilbert spaces.

scholarly journals Ideal Convergence and Completeness of a Normed Space

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 897 ◽  
Author(s):  
Fernando León-Saavedra ◽  
Francisco Javier Pérez-Fernández ◽  
María del Pilar Romero de la Rosa ◽  
Antonio Sala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Convergence Space ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Underlying Space ◽  
Phd Thesis

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

Some sequence spaces and completeness of normed spaces

2017 ◽  
Vol 26 (3) ◽  
pp. 281-287
Author(s):  
RAMAZAN KAMA ◽  
◽  
BILAL ALTAY ◽  
Keyword(s):  
Normed Space ◽  
Summability Method ◽  
Sequence Spaces ◽  
Normed Spaces ◽  
Cesàro Summability ◽  
Cesaro Summability

In this paper we introduce new sequence spaces obtained by series in normed spaces and Cesaro summability method. We prove that completeness ´ and barrelledness of a normed space can be characterized by means of these sequence spaces. Also we establish some inclusion relationships associated with the aforementioned sequence spaces.

scholarly journals Strongly Lacunary Ward Continuity in 2-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hüseyin Çakalli ◽  
Sibel Ersan
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Normed Spaces ◽  
Cauchy Sequences

A functionfdefined on a subsetEof a 2-normed spaceXis strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points inE; that is,(f(xk))is a strongly lacunary quasi-Cauchy sequence whenever (xk) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of continuities are investigated in 2-normed spaces.

scholarly journals Isometries of ultrametric normed spaces

2021 ◽  
Vol 12 (4) ◽  
Author(s):  
Javier Cabello Sánchez ◽  
José Navarro Garmendia
Keyword(s):  
Normed Space ◽  
Normed Spaces ◽  
Tingley’S Problem ◽  
Group Of Isometries

AbstractWe show that the group of isometries of an ultrametric normed space can be seen as a kind of a fractal. Then, we apply this description to study ultrametric counterparts of some classical problems in Archimedean analysis, such as the so called Problème des rotations de Mazur or Tingley’s problem. In particular, it turns out that, in contrast with the case of real normed spaces, isometries between ultrametric normed spaces can be very far from being linear.

scholarly journals Some properties of the superior and inferior semi inner product function associated to the 2-norm

2016 ◽  
Vol 12 (5) ◽  
pp. 6254-6260
Author(s):  
Stela Ceno
Keyword(s):  
Normed Space ◽  
Scalar Product ◽  
Inner Product ◽  
Normed Spaces ◽  
Close Link ◽  
Long Period ◽  
Product Function

Special properties that the scalar product enjoys and its close link with the norm function have raised the interest of researchers from a very long period of time. S.S. Dragomir  presents concrete generalizations of the scalar product functions  in a normed space and deals with the interesting properties of them. Based on S.S. Dragomirs idea in this paper we treat generalizations of superior and inferior scalar product functions in the case of semi-normed spaces and 2-normed spaces.

scholarly journals Some fixed point theorems in n-normed spaces

2020 ◽  
Vol 25 (3) ◽  
pp. 1-15 ◽  
Author(s):  
Hanan Sabah Lazam ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Normed Space ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Weak Contraction ◽  
Contraction Mappings ◽  
Basic Properties ◽  
Definition Of

In this article, we recall the definition of a real n-normed space and some basic properties. fixed point theorems for types of Kannan, Chatterge, Zamfirescu, -Weak contraction and  - (,)-Weak contraction mappings in  Banach spaces.

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